I wonder if that would be productive or counterproductive...

Are you wondering after the productivity of the gift idea in particular or the productivity of the concept as a whole? If it's the gift idea, then I don't see how it wouldn't be productive; a desirous prize would ensure all players apply themselves to the game, and dedicate themselves to learning the skill in the short term. If the game proceeds as expected, then all involved would have had a jolly good time; memories linked to a strong emotion are more memorable (learned this somewhere - no citation I can recall), so the positive fun emotion tied to the game, and thus the skill, will insure its memorability.

The idea as a whole, however, could indeed be quite counterproductive. Though I assume there would be some down time during the mini-camp? walking between sessions, meal times, pre-sleep socializing? The game might fit well during those periods, however it could just as well hinder socialization through added pressure and unnecessary competition.

That particular element seems like it would incentivize campers to spend the period hyper-aware of their own and others' specificity, which seems counterproductive to me. The goal is an increase in the specificity of statements made casually, which could be entirely unrelated. Extending the period to say, a week, might work to prevent this- at that point it would be a long term incentive rather than a prize.